Remark on non-contractible closed geodesics
and homotopy groups

Egor Shelukhin [email protected] Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada and Jun Zhang [email protected] The Institute of Geometry and Physics, University of Science and Technology of China, 96 Jinzhai Road, Hefei Anhui, 230026, China

Abstract.

We prove that if the $m$ -th homotopy group for $m\geq 2$ of a closed manifold has non-trivial invariants or coinvariants under the action of the fundamental group, then there exist infinitely many geometrically distinct closed geodesics for a $C^{4}$ -generic Riemannian metric. If moreover there are infinitely many conjugacy classes in the fundamental group, then the same holds for every Riemannian metric.

1. Introduction and main results

1.1. Introduction

The question of the existence of infinitely many geometrically distinct closed (periodic) geodesics on Riemannian or Finsler manifolds was studied extensively in the past. This question seeks to determine whether or not for every metric or for a class of metrics on a given manifold $M$ there exist infinitely many prime closed geodesics: those closed geodesics that are not obtained as iterations of others.

The efforts were primarily focused on the simply-connected case, where by [14, 32, 18] a manifold with finitely many prime closed geodesics must have ${\mathbb{K}}$ -cohomology, for ${\mathbb{K}}={\mathbb{Q}}$ or ${\mathbb{K}}=\mathbb{F}_{p},$ generated as a unital ${\mathbb{K}}$ -algebra by a single element. Manifolds satisfying this condition for some ${\mathbb{K}}$ do exist: notable examples are the compact rank-one symmetric spaces (CROSS): $S^{n},{\mathbb{C}}P^{n},{\mathbb{R}}P^{n},{\mathbb{H}}P^{n},{\mathbb{O}}P^{2}$ . For such spaces it is known [25] that a $C^{2}$ -generic metric must have infinitely many prime closed geodesics. Prior work [19, 16] proves the same for $C^{4}$ -generic metrics: [19] implies that $C^{4}$ -generically on any closed manifold there are infinitely many prime closed geodesics, unless they are all hyperbolic, while [16] proves that for manifolds rationally homotopy equivalent to a CROSS other than ${\mathbb{R}}P^{n},$ given that all prime closed geodesics are hyperbolic, there are infinitely many of them. The case of ${\mathbb{R}}P^{n}$ follows by a quick covering argument, as does the case of any closed manifold with finite fundamental group, or by the paper [4] discussed below. A celebrated result [5, 12, 17] proves that every Riemannian metric on $S^{2}$ has infinitely many prime closed geodesics, and the same is true for ${\mathbb{R}}P^{2}$ . The analogous statement for any other CROSS is a well-known open question.

The case of infinite fundamental groups at first appears to be simpler, as by a classical theorem often attributed to Cartan or to Hilbert (see for example [11, Chapter 12, Theorem 2.2]), in every non-trivial free homotopy class of loops on $M$ there exists a closed geodesic. However, firstly, it is an open question in group theory [22, 8] whether an infinite finitely presented group could have finitely many conjugacy classes. We recall that the conjugacy classes in $\pi_{1}(M)$ correspond to free homotopy classes of loops on $M$ . Secondly, even if we assume that there are infinitely many conjugacy classes, the simplification appears to be quite illusory, starting with the fact that non-homotopic closed geodesics might not be geometrically distinct (see also [29]). Existence of infinitely many prime closed geodesics is currently known under stronger conditions involving the fundamental group: for example [6] if the manifold is not the circle and its fundamental group is ${\mathbb{Z}}$ (see [30, 31, 26] for other conditions of this kind and [15, 3] for prior work). Generic existence (in $C^{4}$ topology) of infinitely many prime closed geodesics was proved under special conditions on $\pi_{1}(M)$ in [4]: for instance this covers the case where $\pi_{1}(M)$ has finitely many conjugacy classes, yet is not simply-connected. The recent work [26] proves $C^{4}$ -generic existence of infinitely many prime closed geodesics on every closed $3$ -manifold, and produces further results about closed geodesics on connected sums of closed manifolds (see also [23, 21]).

Our main result shows that $C^{4}$ -generic existence of infinitely many prime closed geodesics holds on closed manifolds, of any dimension, under a topological condition regarding the action of $\pi_{1}(M)$ on $\pi_{m}(M)$ for $m\geq 2,$ in the spirit of Bangert-Hingston [6], Albers-Frauenfelder-Oancea [1] and Taimanov [30], which have inspired this paper. Namely, we assume that this action for some $m\geq 2$ has either non-trivial invariants or non-trivial coinvariants. We also prove that if in addition there are infinitely many free homotopy classes of loops on $M,$ then there exist infinitely many prime closed geodesics for every Riemannian metric on $M.$

1.2. Main results

Theorem A.

Let $M$ be a closed manifold. Suppose that there exists $x\in\pi_{m}(M),$ $x\neq 0,$ for $m\geq 2$ such that $a\cdot x=x$ for all $a\in\pi_{1}(M)$ or a homomorphism $\xi:\pi_{m}(M)\to A,$ $\xi\neq 0$ to an abelian group $A$ such that $a^{*}\xi=\xi$ for all $a\in\pi_{1}(M).$ Moreover, let the set $\pi_{1}(M)/{\mathrm{conj}}$ of conjugacy classes in $\pi_{1}(M)$ be infinite. Then for every Riemannian metric on $M$ there exist infinitely many geometrically distinct closed geodesics.

Remark 1.

We remark that our condition on the higher homotopy groups does not constrain the fundamental group of the manifold. For instance, taking a product with the two-sphere does not change the fundamental group but ensures that this condition holds (see also Remark 5). The condition on coinvariants for $m=2$ and $A=GL(1,\mathbb{F}_{p})$ for a finite field $\mathbb{F}_{p}$ is satisfied if and only if there exists a non-trivial rank-one $\mathbb{F}_{p}$ -local system on ${\mathcal{L}}M$ with a trivial restriction to $M$ via the constant loop embedding $M\to{\mathcal{L}}M.$ We refer to [1, Section 2.2] for a related discussion and [1, Proposition 9], showing that this condition, for a suitable prime $p,$ is in turn implied by the non-triviality of the image $H_{2}^{S}(M;{\mathbb{Z}})$ of the second Hurewicz map $\pi_{2}(M)\to H_{2}(M;{\mathbb{Z}}).$

Recall that the natural map $\pi_{1}(M)\to\pi_{0}({\mathcal{L}}M)$ from the fundamental group of $M$ to the set $\pi_{0}({\mathcal{L}}M)$ of connected components of the free loop space ${\mathcal{L}}M$ of $M$ induces a canonical isomorphism of sets

\pi_{1}(M)/{\mathrm{conj}}\xrightarrow{\cong}\pi_{0}({\mathcal{L}}M),

where $\pi_{1}(M)/{\mathrm{conj}}$ is the set of conjugacy classes in $\pi_{1}(M).$ Hence requiring that the set $\pi_{1}(M)/{\mathrm{conj}}$ is infinite is the least possible assumption on the richness of the fundamental group that would help establishing the existence of infinitely many geometrically distinct closed geodesics. In fact, a well-known open question in group theory, due to Makowsky [22] (see also [8]), is whether every infinite finitely-presented group has infinitely many conjugacy classes. Note that the assumption on the conjugacy classes yields infinitely many non-homotopic closed geodesics, as by Cartan and Hilbert’s theorem, there is a closed geodesic in every non-trivial free homotopy class $\alpha$ of loops on $M.$ (It is given as point of global minimum of energy in the corresponding connected component ${\mathcal{L}}_{\alpha}M$ of ${\mathcal{L}}M$ .) However, these geodesics might easily not all be geometrically distinct since a geodesic $c$ and its $k$ -th iterate $c^{k},$ given by $c^{k}(t)=c(kt),$ could contribute to different free homotopy classes. In general, the free homotopy class $[c^{k}]\in\pi_{0}({\mathcal{L}}M)$ of the iterate $c^{k}$ might depend on $k$ in a complicated way.

Corollary 2.

Let $M$ be a closed manifold. Suppose that there exists $x\in\pi_{m}(M),$ $x\neq 0,$ for $m\geq 2$ such that $a\cdot x=x$ for all $a\in\pi_{1}(M)$ or a homomorphism $\xi:\pi_{m}(M)\to A,$ $\xi\neq 0$ to an abelian group $A$ such that $a^{*}\xi=\xi$ for all $a\in\pi_{1}(M).$ Then for a $C^{4}$ -generic Riemannian metric on $M$ there exist infinitely many geometrically distinct closed geodesics.

We remark that in this corollary we make no assumption on conjugacy classes in the fundamental group. It appears to be new even for manifolds of the form $M=X\times S^{2},$ where $X$ is an arbitrary closed manifold (see also Remark 5). Of course $X$ for which prior methods do not apply are more restricted; for instance there should be an element $\alpha\in\pi_{1}(X)/{\mathrm{conj}}$ such that $\alpha^{k},$ $k\geq 1,$ are all distinct.

Proof of Corollary 2.

If $M$ is simply connected, the conclusion is well-known by the work of Klingenberg-Takens [19], Hingston [16], and Rademacher [25]. If $M$ is not simply connected, consider a non-trivial class $a\in\pi_{1}(M).$ Then either there exist two equal conjugacy classes $[a^{k}]=[a^{l}]$ for some positive integers $0<k<l,$ in which case the conclusion follows by work of Ballmann-Thorbergsson-Ziller [4, Theorem A], or the conjugacy classes $[a^{k}],$ for $k$ a positive integer, are all distinct, in which case the conclusion follows by Theorem A. ∎

Remark 3.

By the arguments proving [10, Corollary B], one can upgrade Corollary 2 to the stronger conclusion that for $C^{4}$ -generic metrics, the number of prime periodic orbits of length at most $T$ grows exponentially in $T.$ (Note that while this is not stated explicitly, [10, Corollary B] is proven therein for simply-connected manifolds.)

Remark 4.

It is not hard to see that Theorem A and Corollary 2 extend to Finsler metrics (the latter in the $C^{6}$ -generic case), by a finite-dimensional approximation approach (see [24] as well as [26, 27]). In forthcoming work [28] we will show that an analogue of Corollary 2 holds also for Reeb flows of arbitrary contact forms on the unit cotangent bundle $S^{*}M,$ by proving a weaker analogue of Theorem A. In future work we hope to prove a more direct analogue of Theorem A in the context of Reeb flows.

Remark 5.

It is easy to see that the family ${\mathcal{N}}$ of manifolds satisfying the condition of Corollary 2 is an ideal in the family ${\mathcal{M}}$ of all closed manifolds with respect to the Cartesian product. In certain situations, this property is also preserved under connected sums. For instance, consider the family ${\mathcal{H}}_{2}\subset{\mathcal{M}}$ of all closed manifolds $M$ satisfying $H_{2}^{S}(M;{\mathbb{Z}})\neq 0$ (see Remark 1). Set ${\mathcal{H}}^{\geq m}_{2}={\mathcal{H}}_{2}\cap{\mathcal{M}}^{\geq m},$ where ${\mathcal{M}}^{\geq m}\subset{\mathcal{M}}$ is the subfamily of manifolds of dimension at least $m.$ Let $m=3.$ Then ${\mathcal{H}}^{\geq 3}_{2}$ is an ideal in ${\mathcal{M}}^{\geq 3}$ with respect to connect sum: if $X\in{\mathcal{M}}^{\geq 3}$ and $Y\in{\mathcal{H}}^{\geq 3}_{2}$ then $X\#Y\in{\mathcal{H}}^{\geq 3}_{2}.$ This is a consequence of the fact that for manifolds $X,Y\in{\mathcal{M}}^{\geq 3},$ by the homotopy exact sequence of a pair, $\pi_{2}(X\#Y)\cong\pi_{2}(X)\oplus\pi_{2}(Y),$ and the Mayer-Vietoris sequence for the natural covering $X^{\prime}\cup Y^{\prime}=X\#Y,$ where $X^{\prime}\cong X\setminus B_{X},Y^{\prime}\cong Y\setminus B_{Y}$ for closed balls $B_{X}\subset X,B_{Y}\subset Y.$ In all dimensions $n$ other than $n\in\{0,1,3\}$ there exist simply-connected manifolds $X^{n}\in{\mathcal{H}}_{2}$ of dimension $n.$ Indeed, for $n=2,$ $X^{2}=S^{2}$ and for $n\geq 4,$ we can take $X^{n}=S^{2}\times S^{n-2}.$ Combining with the previous point, we see by the Van Kampen theorem, that for every manifold $M\in{\mathcal{M}}^{\geq 4}$ there exists a manifold $M^{\prime}\in{\mathcal{H}}^{\geq 4}_{2}$ with $\pi_{1}(M^{\prime})\cong\pi_{1}(M).$ We can take $M^{\prime}=M\#X^{n}.$ Similarly, for all $M\in{\mathcal{M}},$ $M^{\prime\prime}=M\times X^{n}\in{\mathcal{H}}_{2}$ satisfies $\pi_{1}(M^{\prime\prime})\cong\pi_{1}(M).$

Remark 6.

One could imagine a generalization of Corollary 2 which would hold for all closed non-aspherical manifolds, and a suitable generalization of Theorem A. However, the complicated algebraic properties of the group rings ${\mathbb{Z}}[\pi_{1}(M)],$ which are, for instance, often non-Noetherian [20], seem to be an obstruction to proving such results.

2. Preliminaries

2.1. Average index and homotopy groups.

Let $(M,g)$ be a Riemannian manifold of dimension $\dim M=n.$ Let $\operatorname{\mathrm{ind}}(c)$ denote the index of a closed geodesic $c$ in $(M,g).$ The average index (or mean-index) of $c$ is defined as

\Delta(c)=\lim_{m\to\infty}\frac{\operatorname{\mathrm{ind}}(c^{m})}{m}\in{\mathbb{R}}.

It satisfies the following useful properties (see [13, Proposition 6.1, Equation (6.1.3)] and [24, Korollar 4.4] for instance):

(1)

$\Delta(c^{m})=m\Delta(c)$ for all $m\in{\mathbb{Z}}_{>0}$
(2)

$|\Delta(c)-\operatorname{\mathrm{ind}}(c)|\leq n-1,$ $|\Delta(c)-(\operatorname{\mathrm{ind}}(c)+\mathrm{null}(c))|\leq n-1$
(3)

$\Delta(c)\geq 0$ and $\Delta(c)=0$ if and only if $\operatorname{\mathrm{ind}}(c^{m})=0$ for all $m\in{\mathbb{Z}}_{>0}.$

Let $\Lambda_{\alpha}$ be the $L^{2}_{1}$ -completion of ${\mathcal{L}}_{\alpha}M$ and $\Lambda_{\alpha}^{A}=\{\mathcal{A}<A\}$ for $A>0.$ Note that the inclusion ${\mathcal{L}}^{A}_{\alpha}M\to\Lambda^{A}_{\alpha}M$ is a homotopy equivalence for every $A>0$ by a result of Anosov [2] (see also [13, Proposition 2.2]).

We require the following property of the homotopy groups of $\Lambda_{\alpha}M,$ which follows from [14] and the properties of the mean-index (see also [9]).

Proposition 7.

If all geodesics $b\in\Lambda_{\alpha}M$ with $\mathcal{A}(b)\geq A$ are of mean-index $\Delta(b)>k+n-1,$ then $\pi_{m}(\Lambda_{\alpha})=\pi_{m}(\Lambda_{\alpha}^{A})$ for all $m\leq k.$

We recall the following long exact sequence from [30]. Fix $\alpha\in\pi_{0}({\mathcal{L}}M).$ Consider the evaluation map $ev:\Lambda_{\alpha}M\to M,$ $z\mapsto z(0),$ where $z\in\Lambda_{\alpha}M$ is considered as an absolutely continuous map $z:{\mathbb{R}}/{\mathbb{Z}}\to M$ with square-integrable velocity vector. This map is a Serre fibration. Fix base-points $x_{0}\in M$ and $\gamma_{0}\in ev^{-1}(x_{0}).$ Let $a\in\pi_{1}(M,x_{0})$ be the class represented by $\gamma_{0}.$ Of course $[a]=\alpha$ in $\pi_{1}(M)/{\mathrm{conj}}.$

Proposition 8.

The long exact sequence of a fibration for $ev$ and an identification $\pi_{k}(ev^{-1}(x_{0}),\gamma_{0})\cong\pi_{k+1}(M,x_{0})$ yield the long exact sequence:

\ldots\to\pi_{m+1}(M,x_{0})\xrightarrow{1-a}\pi_{m+1}(M,x_{0})\to\pi_{m}(\Lambda_{\alpha}M,\gamma_{0})\to\pi_{m}(M,x_{0})\xrightarrow{1-a}\pi_{m}(M,x_{0})\to\ldots,

for $m\geq 2,$ where $a:\pi_{k}(M,x_{0})\to\pi_{k}(M,x_{0})$ is the standard action of the fundamental group on higher homotopy groups. For $m=1$ the sequence reads:

\ldots\to\pi_{2}(M,x_{0})\xrightarrow{1-a}\pi_{2}(M,x_{0})\to\pi_{1}(\Lambda_{\alpha}M,\gamma_{0})\to C_{a}\to 1,

where $C_{a}\subset\pi_{1}(M,x_{0})$ is the centralizer of $a.$

3. Proof of main result

Proof of Theorem A.

By Proposition 8, the hypothesis of the theorem implies that $\pi_{\ell}(\Lambda_{\alpha}M)\neq 0$ for $\ell=m$ or $\ell=m-1$ and all $\alpha\in\pi_{0}({\mathcal{L}}M).$

Suppose by contradiction that there are finitely many prime closed geodesics. (In particular, every closed geodesic is isolated.) We denote them by $\gamma_{1},...,\gamma_{P},\gamma_{P+1},...,\gamma_{N},$ such that the geodesics $\gamma_{1},...,\gamma_{P}$ have mean-index $\Delta(\gamma_{i})>0$ and $\gamma_{i}$ for $i>P$ have $\Delta(\gamma_{i})=0.$ Then by Property (3) of the mean-index, for every $i>P,$ $\operatorname{\mathrm{ind}}(\gamma_{i}^{j})=0$ for all iterations $j.$ Now all closed geodesics are iterations of $\gamma_{1},\ldots,\gamma_{N}$ . Consider the set $S$ of all the conjugacy classes $[\gamma_{i}^{j}],$ $1\leq i\leq P$ such that $\Delta(\gamma_{i}^{j})\leq n-1+\ell.$ Since the mean-indices of $\gamma_{i}$ for $1\leq i\leq P$ are positive, by Property (1) of the mean-index, $S$ is a finite set. Since $\pi_{0}({\mathcal{L}}M)$ is an infinite set, so is the complement $\pi_{0}({\mathcal{L}}M)\setminus S.$ Consider $\alpha\in\pi_{0}({\mathcal{L}}M)\setminus S.$ Now let $c_{1},\ldots,c_{r}\in\Lambda_{\alpha}M$ be all the points of absolute minimum of the energy functional on $\Lambda_{\alpha}M$ . Then the level set of this minimum is the disjoint union $\sqcup_{j=1}^{r}S(c_{j})),$ where $S(c_{j})$ is the circle given by $c_{j}$ and its rotations, and therefore its $k$ -th homotopy groups vanish for all $k>1.$ We now claim that there exists a geodesic $\eta\in\Lambda_{\alpha}M$ that is not a global minimum and is not an iteration of any of $\gamma_{1},...,\gamma_{P}.$ Note that as $\eta$ must be an iteration of some $\gamma_{i}$ for $i>P$ we must have $\operatorname{\mathrm{ind}}(\eta^{r})=0$ for all $r\geq 1.$ By Bangert-Klingenberg [7, Theorem 3], this implies that there exist infinitely many prime closed geodesics on $M.$

To prove the last claim, we treat the cases $\ell\geq 2$ and $\ell=1$ slightly differently. Suppose first that $\ell\geq 2.$ The contrapositive assumption that all closed geodesics above the minimum energy are iterations of $\gamma_{1},\ldots,\gamma_{P}$ , implies that $\pi_{\ell}(\Lambda_{\alpha}M)=0$ by Proposition 7, since by construction, any iteration $\eta$ of $\gamma_{1},\ldots,\gamma_{P}$ in $\alpha$ has $\Delta(\eta)>n-1+\ell.$ But we know that $\pi_{\ell}(\Lambda_{\alpha}M)\neq 0.$ In the case where $\ell=1,$ we observe that the coinvariant hypothesis and Proposition 8 yields that $\ker(ev:\pi_{1}(\Lambda_{\alpha}M,\gamma_{0})\to\pi_{1}(M,x_{0}))\neq 0.$ Now, we argue that there can only be one point of global minimum under the contrapositive assumption. Indeed, had we $N>1$ global minima $c_{1},\ldots,c_{N}$ , the contrapositive assumption would yield that $\pi_{0}(\Lambda_{\alpha}M)$ is a set of $N$ elements, which contradicts the connectedness of $\Lambda_{\alpha}M.$ Hence there is a unique global minimum $c.$ Now the contrapositive assumption yields that $\pi_{1}(\Lambda_{\alpha}M)\cong\pi_{1}(S(c))\cong{\mathbb{Z}}$ and the evaluation map $\pi_{1}(S(c))\to C_{a}$ is injective. This is a contradiction to its kernel being non-zero. This proves the claim for all $\ell\geq 1.$ ∎

Acknowledgements

We thank the participants of the reading group on closed geodesics at the Université de Montréal, especially Marco Mazzucchelli, for useful discussions. E.S. was supported by an NSERC Discovery grant, by the Fonds de recherche du Québec - Nature et technologies, by the Fondation Courtois, and by an Alfred P. Sloan Research Fellowship. This work was partially supported by the National Science Foundation under Grant No. DMS-1928930, while E.S. was in residence at the Simons Laufer Mathematical Sciences Institute (previously known as MSRI) Berkeley, California during the Fall 2022 semester. J.Z. is supported by USTC Research Funds of the Double First-Class Initiative. This work was initiated when J.Z. was a CRM-ISM Postdoctoral Research Fellow at CRM, Université de Montréal, and he thanks this institute for its warm hospitality.

References

[1] P. Albers, U. Frauenfelder, and A. Oancea. Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity. Math. Ann., 367(3-4):1403–1428, 2017.
[2] D. V. Anosov. Some homotopies in a space of closed curves. Izv. Akad. Nauk SSSR Ser. Mat., 44(6):1219–1254, 38, 1980.
[3] W. Ballmann. Geschlossene Geodätische auf Mannigfaltigkeiten mit unendlicher Fundamentalgruppe. Topology, 25(1):55–69, 1986.
[4] W. Ballmann, G. Thorbergsson, and W. Ziller. Closed geodesics and the fundamental group. Duke Math. J., 48(3):585–588, 1981.
[5] V. Bangert. On the existence of closed geodesics on two-spheres. Internat. J. Math., 4(1):1–10, 1993.
[6] V. Bangert and N. Hingston. Closed geodesics on manifolds with infinite abelian fundamental group. J. Differential Geom., 19(2):277–282, 1984.
[7] V. Bangert and W. Klingenberg. Homology generated by iterated closed geodesics. Topology, 22(4):379–388, 1983.
[8] G. Baumslag, A. G. Myasnikov, and V. Shpilrain. Open problems in combinatorial group theory. Second edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), volume 296 of Contemp. Math., pages 1–38. Amer. Math. Soc., Providence, RI, 2002.
[9] K.-c. Chang. Infinite-dimensional Morse theory and multiple solution problems, volume 6 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 1993.
[10] G. Contreras. Geodesic flows with positive topological entropy, twist maps and hyperbolicity. Ann. of Math. (2), 172(2):761–808, 2010.
[11] M. P. a. do Carmo. Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty.
[12] J. Franks. Geodesics on $S^{2}$ and periodic points of annulus homeomorphisms. Invent. Math., 108(2):403–418, 1992.
[13] M. Goresky and N. Hingston. Loop products and closed geodesics. Duke Math. J., 150(1):117–209, 2009.
[14] D. Gromoll and W. Meyer. Periodic geodesics on compact riemannian manifolds. J. Differential Geometry, 3:493–510, 1969.
[15] M. Gromov. Three remarks on geodesic dynamics and fundamental group. Enseign. Math. (2), 46(3-4):391–402, 2000.
[16] N. Hingston. Equivariant Morse theory and closed geodesics. J. Differential Geom., 19(1):85–116, 1984.
[17] N. Hingston. On the growth of the number of closed geodesics on the two-sphere. Internat. Math. Res. Notices, (9):253–262, 1993.
[18] J. Jones and J. McCleary. String homology, and closed geodesics on manifolds which are elliptic spaces. Algebr. Geom. Topol., 16(5):2677–2690, 2016.
[19] W. Klingenberg and F. Takens. Generic properties of geodesic flows. Math. Ann., 197:323–334, 1972.
[20] P. Kropholler and K. Lorensen. Group-graded rings satisfying the strong rank condition. J. Algebra, 539:326–338, 2019.
[21] P. Lambrechts. The Betti numbers of the free loop space of a connected sum. J. London Math. Soc. (2), 64(1):205–228, 2001.
[22] J. A. Makowsky. On some conjectures connected with complete sentences. Fund. Math., 81:193–202, 1974.
[23] G. P. Paternain and J. Petean. On the growth rate of contractible closed geodesics on reducible manifolds. In Geometry and dynamics, volume 389 of Contemp. Math., pages 191–196. Amer. Math. Soc., Providence, RI, 2005.
[24] H.-B. Rademacher. Morse-Theorie und geschlossene Geodätische, volume 229 of Bonner Mathematische Schriften [Bonn Mathematical Publications]. Universität Bonn, Mathematisches Institut, Bonn, 1992. Habilitationsschrift, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1991.
[25] H.-B. Rademacher. On a generic property of geodesic flows. Math. Ann., 298(1):101–116, 1994.
[26] H.-B. Rademacher and I. A. Taimanov. Closed geodesics on connected sums and 3-manifolds. J. Differential Geom., 120(3):557–573, 2022.
[27] H.-B. Rademacher and I. A. Taimanov. The second closed geodesic, the fundamental group, and generic Finsler metrics. Math. Z., 302(1):629–640, 2022.
[28] E. Shelukhin, V. Stojisavljevic̀, and J. Zhang. Work in progress.
[29] I. A. Taimanov. The type numbers of closed geodesics. Regul. Chaotic Dyn., 15(1):84–100, 2010.
[30] I. A. Taĭmanov. Closed geodesics on non-simply-connected manifolds. Uspekhi Mat. Nauk, 40(6(246)):157–158, 1985.
[31] I. A. Taĭmanov. Closed geodesics on non-simply connected manifolds. Sibirsk. Mat. Zh., 34(6):170–178, iv, ix, 1993.
[32] M. Vigué-Poirrier and D. Sullivan. The homology theory of the closed geodesic problem. J. Differential Geometry, 11(4):633–644, 1976.

Remark on non-contractible closed geodesics and homotopy groups